a gymnasiou 14 15 a tomos
DESCRIPTION
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( ) (+1,2) (+2,6) ( ) 4,9 ( 1,9) ( ) ( 11) (+4,2) ( ) 1,5
( ) 15 14 ( ) 2
3. 3 :
( ) 10, 8, 6, 4, , , ( ) 10, 7, 4, 1, , ,
( ) 19, 14, 9, 4, , , ( ) 1 , 3, 4 , 6 , , ,
4. >, = , < ,
:
( ) 2000 . . . . . ( 2000) ( ) (+5) . . . . . (+5) ( 10)
( ) 2 + 3 . . . . . 2 4 ( ) 4 . . . . . 4 ( ) | 1| | 2| . . . . . ( 1) ( 2) ( ) 3 ( 2) . . . . . | 1 + 2|
148
4: –
5. – N . 1962
235 , 2002 182 .
2002 1962.
6. , :
( ) ( 5) = +1 ( ) (+11) = +4
( ) 5 + = 13 ( ) +12 = 12
( ) | 11| + = +20 ( ) | 3 | = +10 ( ) + = ( ) 1 + = 2
( ) 2 + > 0 ( ) 7 > 0
7. :
( ) (+10) + ( 15) + ( 3) ( 1) (+7)
( ) ( 7) + ( 5) (+8) + ( 7) ( 15)
( ) 12 (11 3) + (5 7) (8 + 6)
( ) (13,7 2,6) + 14,8 ( 8,7 + 5)
( ) +
8. = 3 = + , :
= 3 + = | |
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. , .
( ) =
( ) =
( ) =
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149
4: –
10. . , ,
.
= ( 2,3 + 7,2) ( 5,9) = ( 104,99 4,8) ( 100 + 2,99) ( 100,3)
11. + ,
:
12. . . .. ,
, .
, 08: 00 ,
08: 00. ( )
. ( ) ,
17: 00;
150
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829,84 160 57
504 18
, , 5 .
, ,
. :
. : 4 ; 3 ;
, 20 ; , 45 ;
,
. : 4 ; 3 ;
, 10 ; , 40 ;
«A_En_4_Ginomeno_1
.
151
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.
. 0 .
.
. .
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6. .
152
4: –
+ : (+3) (+7) = +(7 3) = +21 ( 8)( 1) = +(8 1) = +8
: (+3)( 7) = (7 3) = 21 , , = ( ) = ( ) ( + ) = + ( ) = .
: ( 2)(+3) = (+3)( 2) [( 2)(+7)]( 1) = ( 2)[(+7)( 1)] ( 2)[( 4) + (+6)] = ( 2)( 4) + ( 2)( 6) = =
: ( 3)(+1) = ( 3) : ( 3)( 1)( 2) = 6 ( 1)( 5)( 4)( 2) = +40
(+3) (+7) (+3)(+7) ---
153
4: –
0
=
:
4 .
1. : ( ) 3 ( 4) ( ) 2 (+2) ( ) 2 ( 3) ( ) 0 ( 2014) ( )
( ) 3 ( 4)
= (3 4) = 12
. , .
( ) 2 (+2)
= (2 2) = 4
. , .
( ) 2 ( 3)
= +6 . , .
( ) = =
. , .
( ) 0 ( 2014) = 0 . , .
( ) = + = +1
. , +1.
154
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2. 1000 .
30 000 , : ( ) 3 ; ( ) 3 ;
( ) 1000 , 3 : ( 3)( 1000) = +3000
, 30000 + 3000 = 33000 .
( ) 1000 , 3 : (+3)( 1000) = 3000. , 30000 3000 = 27000 .
3. , = 0,5. = +
= + = 0,5 0,5 + 0,5 = 0,5 (0,5 + 0,5) (0,5 0,5) = 0,5 1 0 = 0
-.
- -
.
4. : = ( 2)(+3)( 4)(+5) + ( 2)( 3)( 4)(+5)
. 2
. 3 . = ( 2)(+3)( 4)(+5) + ( 2)( 3)( 4)(+5)
= +(2 3 4 5) (2 3 4 5) = 0
, . , .
5. :
( ) 2( + 5) ( ) (4 6)
155
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- , -
- .
( ) (4 6) = 4 ( 6) = 2 + 3 -
.
1. : ( ) 2 ( )
( ) 2 ( ) 3,2 5,1
2. : ( ) ( 7) (+2) ( ) ( 2) ( 5)
( ) ( 3) 0 ( ) 2 (+3) 3 ( 12) ( ) 5 ( 11)
( ) ( 6,2) ( 3) ( ) ( 1) 0
( ) 8 ( ) ( 2,1) ( 3,2)( ) ( ) 1
3. :
( ) (+3) ( 1) (+2) ( ) (+3) ( 1) ( 2)
( ) (+3) ( 1) ( 2) ( 2) ( ) (+3) (+1) (+2) ( 1)
( ) (+3) 0 (+2,5) (+1,2) ( ) ( 1) ( 1) … ( 1)
( ) + ( ) ( 1) (+1) ( 1) (+1) … (+1)
( ) 1 0 ( ) (+1) ( 1) (+1) ( 1) … (+1)
4. , :
+3 2,5 3 +7 1 +
156
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5. .
.
6. ,
:
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/
( ) - -
.
/
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/
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/
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/
7. :
( ) 3 ( 6 + 8) ( ) 3 + 6 ( 2)
( ) 6 2 (5 + 4) ( ) 6 ( 2) ( 8)
( ) 2 ( 4) ( 1) ( 1) ( ) 3 ( 6) + 3 ( 8 + 2)
( ) 5 ( 4 + 3) 2 21 ( ) 2 ( 6 + 10) ( 15)
( ) ( 3 + 3) ( 3) ( 3) ( ) ( 1) ( 1) … ( 1) ( 1)
8. , :
( ) ( 3) = ( ) + + = ( ) = ( ) 2 = +1 ( ) ( 5) = 0 ( ) +2 = 2 (+1)( ) ( 3) = 1 ( ) ( 2012) = ( 1) ( 2)
( ) 1, 2, 4, 8, , , ( ) 2, 2, 2, 2, , ,
( ) 1, 10, 100, 1000, , , ( ) 1, 3, 9, 27, , ,
157
4: –
9. 8 6 .
10. 2,5 , +9 , 0 , 1 , +10 , 2100. :
( ) , ( ) .
11. , = 1. = 2 + ( + 1) = (3 + 2)(1 )( + 0,5) = ( 2010)( 2011)( 2012) + ( 2012)( 2011)(+2010) = ( 2)(+3)(+5) + ( 2)( 3)( 5)
12. :
( ) 3( + 2) ( ) 3( 1) + 5( ) (6 + 32) ( ) 6 ( 5) + 7
( ) 5(3 2) + 7 ( ) (4 + ) ( + 2)
( ) 4( + 7) 6(4 ) ( ) 5( 11) + 2
13. 350 €6 €9.
( ) €6, :
i. €9 ,
ii. €6, iii. €9, iv. .
( ) 120 €6, .
14. := 2( 3 ) + 4(2 ) 2(3 2 ).
, + = 4.
158
4: –
2 . ;
1 2 3 4 (+3) 3 ( 3) (+3) 9 ( 3) (+3) 27 ( 3) (+3) 81 ( 3) (+3) 243 ( 3) (+3) 729 ( 3) (+3) 2187 ( 3) (+3) 6561 ( 3) (+3) 19683 ( 3) (+3) 59049 ( 3) (+3) 14348907 ( 3) (+3) 43046721 ( 3) (+3) 3486784401 ( 3)
; , .
;
3 ;
159
4: –
A > , > .
: (+7) = +343
, . < >
: ( 3) = +9 , . < < .
: ( 2) = 8
1. : ( ) 5
( ) ( 5) ( ) (+4) ( ) ( 1256)
( ) 5 , (5) .
( ) ( 5) , ( 5) (2)
.
( ) (+4) , (+4) .
( ) ( 1256) , ( 1256) (27) .
2. : ( ) ( 4) ( ) 4 ( ) ( 3)
( ) (+4,2) ( ) (+1) ( ) ( 1)
160
4: –
( ) ( 4) = +(4 ) = 16
( ) 4 = (4) = 16
( ) ( 3) = 27( ) (+4,2) = 1( ) (+1) = +1
( ) ( 1) = +1
1. : ( 2)
6,3 2
2. :
3. :
( ) ( 1) = 1 ( ) ( 2) = 16 ( ) (+3) = 27 ( ) ( 1) = 1 ( ) 2 = 16 ( ) 6 = 36 ( ) = ( ) ( 2,7) = 2,7 ( ) (0,5) = 0,125
4. :
( ) ( 5) ( ) 2 ( ) ( )
( ) ( 1) ( ) ( 18)
( ) ( 3) ( ) ( 10)
5.
: ( ) 25 ( ) 8 ( ) 1
( ) ( 3) ( ) ( ) 2012 ( ) (+2,15)
161
4: –
6. < , =, > . .
( ) ( 2) = 8 ( ) ( 5) = 5
( ) 3 = +9 ( ) ( 1) = 1
( ) ( 5) = +25 ( ) ( 1) = 1
( ) 3 = 1 ( ) =
( ) (3 5) > 0 ( ) ( 5 ) < 0
7. < , , > ,
:
( ) (+2) … … ( 2) ( ) ( 7) … … (+7)
( ) (+12) … … ( 12) ( ) (+113) … … ( 113)
( ) ( 2) … … (+2) ( ) (+4) … … ( 4)
8. ( 6) , 6 , ( 6) , 6
, .
9.
.
.
.
. .
10. = 1, : = 2 4 + 1 5
11. :
= ( 1) + ( 1) + ( 1) + + ( 1) + ( 1)
162
4: –
(1)
, .
, 2 .
12 , ;
08: 00 2 ,
8 ;
03: 00, 08: 00 2 ;
(2)
12 15.
.
.
163
4: –
.
+ : (+12) (+2) = +(12 2) = +6 ( 30) ( 5) = +(30 5) = +6
: ( 45) (+5) = (45 5) = 9 (+2,1) ( 7) = (2,1 7) = 0,3 = , 0
: ( 5) (+2) = ( 5) + = = = =
: = = = = = 5 – = = 0 , 0, 0
: = = =
--
164
4: –
. , , 0
( + ) = + ( ) =
: [( 4) + (+6)] ( 2) = ( 4) ( 2) + (+6) ( 2)
1. : ( ) 12 3 ( ) 24 ( 3) ( )
( ) ( )
( ) 12 3 = (12 3) = 4 . , .
( ) 24 ( 3) = +(24 3) = 8 . ,
.
( ) = = 5 . ,
.
( ) = = +6 -
. . , .
( ) = · = = 2
, -
.
2. N 85 .
, = ( ) C ,
.
= , 0 = = , 0
165
4: –
= ( ) = ( ) 85. = ( ) .
= .
= 65 .
3. : ( ) ( 3 + 1 + 2) ( 2)
( ) [ 52 + ( 6) ] ( 1) (8 9)
( ) .
( 3 + 1 + 2) ( 2) . = 0 ( 2) -
. = (+1) 0 = +1
.
( ) ( )
. [ 52 + ( 6) ] ( 1) (8 9) = [ 52 + 36] ( 1) ( 1)= ( 16) ( 1) ( 1) = (+16) ( 1) = +16 + 1 = +17
. .
1. := = 20 5 = ( 4) ( 3) = ( 2) (+5) = 8 ( 10) = ( 12) ( 16)
166
4: –
2. : ( ) ( 10) ( 2) ( ) ( 15) 5 ( ) 18 ( 3)
( ) ( 70) ( 7) ( ) ( 2,1) ( 3) ( ) ( 4,5) 9
( ) 50 ( 0,5) ( ) ( 12,24) ( 4) ( ) 80 ( 2)
( ) ( ) 1 : 5 ( ) ( 7) 3. 20 15.
4. +2 +4.
5. <, =, >,
: ( ) (+6) ( 2) . . . . . +5 2 ( ) ( 12) ( 3) . . . . . 12 ( 3)
( ) (+6) . . . . . 0 ( ) ( 1) . . . . . ( 1) ( 1)
( ) (+2) . . . . . 2 2 ( ) (+6) . . . . . + ( 10) 6. ,
: ( ) (+32) = (+8) ( ) ( 24) ( 6) = ( ) ( 25) = +5 ( ) (+24) = +2
( ) ( 3) = 4 ( ) ( 5 ) = 5
( ) ( 4) = ( ) (+3) = 9
( ) = ( ) = 2
( ) = +2 ( ) = 2
7.
.
= ( )
.
49 58 50 41 40 20
167
4: –
8. : ( ) +1 ( ) 1 ( ) 0
9. :
( ) ( 12 4) ( 4) ( ) ( 9 + 3) ( 2 4) ( ) ( 2 + 3 1) ( 3 + 2) ( ) ( 2 5 1) ( 3 + 1)
( ) + + ( ) : ( 4 2)
( ) ( )
10. :
( ) ( 20) 2 ( 2) ( ) ( 21) + 6 3 ( ) 10 + 3 2 7 ( ) / ,
5.
11. , :
( ) ( 10). . . . . ( 2). . . . . (+1) = 21
( ) ( 5). . . . . ( 2). . . . . (+4) = 1
( ) 6. . . . . ( 7). . . . . (+2) = 44
( ) ( 8). . . . . ( 2). . . . . ( 2 ) = 9
12. :
( ) 45 35 ( 5)
( ) 15 ( 2) ( 1 + 3)
( ) ( 2) ( 3) ( 12) ( 1)
( ) 127 25 ( 1 + 2)
( ) 2 [ 3 2 ( 1 1)] ( ) 2 + 2 + ( 1)
( ) ( )( ) ( ) ( )( )( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
168
4: –
13. , = 12, = 4 = 6:
( ) ( )
( ) ( + ) ( + 2) ( ) ( )( + )
( ) ( ) ( )
14. , 11: ( 5) + (+4) ( 2) ( 1) ( 3)
15. , .
.
16. = 2 = 3, :
( ) 3 + 2 + ( ) ( ) ( 1) +
17. : ( ) 3( 3) ( ) 6 ( 2 + 1) ( ) ( 4) + 3 1 (+2) ( ) (2 3) ( 5) + (3 5) ( ) ( 7) ( 7) + ( 3) + 5 1
( ) 7,25 ( 1 + 2) + ( 3) ( 4,2)
( ) ( 2) ( 2) ( 5) +2 ( 3)
( ) ( ) ( )( )
169
4: –
1, , , … , .
1 1 0,111111111111 …. 0,5 0,1
0,333333333333 … 0,090909090909 …
0,25 0,833333333333 …
0,2 0,076923076923 … 0,1666666666666 … 0,0714285714285714 …
0,142857142857 … 0,066666666666666 … 0,125 0,625
.
,
2 / 5, .
.
170
4: –
: 0,333 … = 0, 3 , 3 1. 34,23181818 … = 34,2318 , 18 2.
: 0, 3 34,2318
: = 0, 3125 , = 0, 416 ( 10 = ( , , 0).
= 2, 171
10 3, 172
-
171
4: –
1. : ( ) 2 ( ) ( )
( ) 2 2 = 2 = 2,6.
( ) 5 8
, = 0,625
( ) ,
5 3 .
5,000 3
, = = 1,666 = 1, 6 3 1,666 20 18 20 18 20
2. 0, 3 .
= 0,333 … . 10 = 3,333 … 10. 10 = 3,333 … 0,333 … 9 = 3 =
-
. 0, 3 =
5,000 8 48 0,625 20 16 40 40 0
172
4: –
3. 0,586 . = 0,58666 … . 100 = 58,666 … 100 -
. 1000 = 586,666 … 1000.
1000 100 = 586,666 … 58,666 … 900 = 528 = =
-.
0,586 =
1. :
( ) 4, 7 ( ) 2, 3
( ) 3,417 ( ) 0,28
2. 1 , . 3. :
( ) 1, 1 1,11111111
( ) 3,68 3,68
( ) 4,912 4,9102
4. :
0,69 0, 69 0,069 0, 6 0,69
173
4: –
;
13 10
174
4: –
.
175
4: –
: 2 6 = 3 2 . 6 3 .
:
+ 3 = 5 = 2. 2 1 = 3 = 2. + 3 = 5 2 1 = 3 .
, .
.
+ = + = =
+ =
2 8 = +15 3
= 2
: 2 + 3 = 8 + 15 =
-
, ( ).
.
176
4: –
1. 4 + 3 = 5 5 3 4 + 3 = 5 5 3
.
4 + 3 = 5 5 3 4 5 + 3 = 5 3 2 = 8 . = . = = 4 .
2. : + 1 =
+ 1 =+ =
[3,2,1] = 6 - . = ( )
-. 4 + 6 = 3 2( + 3) .4 + 6 = 3 2 6
.4 3 + 2 = 6 6
. 3 = 12 33 = 123 . = 4 .
177
4: –
3. 30 5 10 .
( ) .
( ) €2,40, .
( ) 5 ,
10 30 .
5 : 10 : 30
, 5 : 5
10 : 10(30 ) = 5 + 10(30 ) = 5 + 300 10 = 300 5
( ) = 240 300 5 = 240
5 = 240 300 5 = 60 = = 12
, 5 12, 10 18. 4. ,
= . 3
, 7
. , 38 .
= . , 2 3,
3(2 3) 7. = = = = 2 3 = = 3(2 3) 7 = 6 9 7 = 6 16
178
4: –
= + + + + = 38 + 6 16 + + 2 3 + 2 3 = 38 + 6 + + 2 + 2 = 38 + 16 + 3 + 3 12 = 60 1212 = 6012 = 5
, = = 5 = = 6 16 = 14
1. . ,
( ).
( )
( )
179
4: –
2. : ( ) 7 + = 5 ( ) 4 = 16
( ) 6 = 3 ( ) 3 4 = 19
( ) 11 = 4 1 ( ) 5 + 2 = 9 2
( ) 8 4 + 3 + 26 = 3 4 + 2 ( ) 2 + 4 = 2 + 4
3. : ( ) 5( + 1) = 3(3 1)
( ) + (5 7) = 5
( ) 5( 2) 2( 1) = 11 ( ) 4( 3) 5 = 0 ( ) 7( 3) 12 = 13
( ) 3( + 2) ( 2) = 4( + 1)
4. :
( ) = ( ) =
( ) = 1 ( ) =
( ) = ( ) =
( ) 1 = + ( ) = 2
5.
:
( ) 3 9 = 18 ( ) 2 = 6 ( ) 5 = 9 3 = 9 = 6 + 2 = 9 5 = 3 = 8 = 4
6. , 10.
7. , , = 5 + 9, = 7 + 5. .
180
4: –
8. , 3
1 . 86 , .
9. 280 . 20
, ,
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11. €5 .
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12.
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.
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€6. 10 . €260,
. 14. 50 2
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181
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1. : ( ) ( 2) ( 3) ( ) ( 2) + ( 3)
( ) ( 2) ( 3) ( ) 2 3
( ) ( 3) ( 5) + (+4) ( ) 3 + 4 | 5| ( ) ( 3) | 5 + 4| ( ) ( 3) ( 5) (+4) ( ) ( 3 5) ( 4) ( ) ( 5) ( 4 + 3)
2. :
( ) (+1) = 1 ( ) (+1) = 1 ( ) ( 2) = 64
( ) = ( ) ( 25) = 25 ( ) ( 4) = 64
3. ,
:
( ) 10 + /
( ) 10 10 /
( ) 10 < 0 /
( ) | 10| < 0 /
( ) | 10| = ( 10) /
( ) 10 < | 10| /
( ) ( 10) > ( 10) /
4. :
( ) ( 3) ( ) (+1) ( ) 2 ( ) ( 2) ( ) ( ) ( 2) ( ) ( 3 + 2) ( ) ( 6 + 1) ( ) ( ) +
5. :
, , ,
182
4: –
6. <, =, >, :
( ) 0 . . . . . 1 ( ) 5 . . . . . 6
( ) 5 . . . . . | 5| ( ) 5 . . . . . 12
( ) +5 . . . . . 5 ( ) 3 . . . . . 3
( ) 2010 . . . . . 2011 ( ) 3 2 1 . . . . . (+3)
( ) 9 + 2 . . . . . ( 1) ( ) ( 3 + 1) . . . . . ( 16) ( 4)
( ) 3 ( 2). . . . . 3 ( 2) ( ) 0 ( 1) . . . . . | 1| 1
7.
:
8. ,
: ( ) (+14) + = +5 ( ) ( 2,4) = 9,2
( ) ( 20) = +100 ( ) ( 8) = 1
( ) ( 2 1) = 1 ( ) (+2) = 4
( ) 12 + = 7 ( ) +2 = +7
( ) (+3 1) = 6 ( ) (+2 4) = | 2| ( ) = 2 + 3 ( ) = 2014 + 2014
9. , .
.( )
. ( ) > , | | > | |. ( ) , | | = | | | |. ( ) , | | = . ( ) .
1 5
5 1 0 3
183
4: –
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11. 48
28 .
, . .
12. :
13. ,
.
: .
( ) :
i. 2 , +4 ii. 2 , 4 iii. +2 , 4 iv. +2 , +4
( ) ; ;
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( ) 4 > > 6 ( ) | | = 12 ( ) | | < 3
( ) ( 1 3) (4 7) ( ) (12 24) ( 3)
( ) ( 6) ( 3) + ( 4) ( ) ( 5) 4 ( 3)
( ) 8 ( 7) 8 7 ( ) 11 4 + ( 8) ( 3)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( )( ) ( ) ( )
184
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15. : ( ) ( 1) 1
( ) : 1( ) ( 2) (+1) 7
( ) ( 1) 2 (+5 1)( ) ( 2) ( 2) ( 3) ( 3 + 4)
( ) ( 4 + 2) 4 [ (+5) ( 3) ] + ( 2)
16. , .
( ) , , .
/
( ) ( 0).
/
( ) .
/
( ) 0.
/
( ) 1.
/
( ) .
/
( ) .
/
17. :
. . i. = 0
. . ii. < 0
. iii. > 0
. . iv. = 1
v. + = 0
185
4: –
18. < , = , >, . < 0, > 0 > 0 , :
( ) | | … … 0 ( ) … … 0
( ) … … 0
( ) … … 0 ( ) … … 0 ( ) … … 0
19. , ( , 0) :
( ) < 0,
:
( ) > 2, :
( ) ( 2) = 0, :
( ) ( )( )( ) > 0, :
20. o , : ( 1) + ( 1) + ( 1) + ( 1) = 0
21. = 3, : = 2 ( 5) 4 ( 2) 4 + 6 = + 5(3 ) 4 4(5 2 )
22. + = 5 = 3,
: ( ) 3 + ( 3 ) ( ) 5 ( ) + 4
23. : ( ) + 6 5 = 14 2 ( ) 16 = 14
( ) 2(y 1) = 3y + 4 ( ) ( 1) = 2
( ) 3 (2 + 1) = 1 ( ) 2 3(2 ) = 4 8
( ) = 2 ( ) =
( ) + = 1 + ( ) + = 2
186
4: –
24. , .
25. 27 . 3
. .
26. 115 ( , ).
5 , , .
27. €160.
, €20 .
;
28. 50 €1200.
€20 . .
29. : ,
, 100 . 5 50,
100 . ;
30. . €20 , .
;
187
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+ ( 2) 2. , , , – > 0, = 222 < 0,
.
3. | + | = | | + | |.
.
4. . 100 7 , 0 3
. ; ,
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5. : ( 200 + 199) ( 199 + 198) + ( 198 + 197) ( 197 + 196) + + ( 2 + 1) ( 1) 6. , 34
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.
.
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7. , : |2010 | = 2010 > 0
188
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8. ,
.
.
9. 4 ( + 3) = ( 2) 7 + 5 = 2 + 3, .
10. , :
( 1) + (+1) ( 1) + 1 + ( ) + ( )
11. = 1 = 2, : = 4 + 3( + ) + 3(3 + )
12.
, ier in i:
190
:
191
:
11
1. ( ) 258, 285, 528, 582, 825, 852
( ) 258, 285, 528, 582, 825, 852
( ) : 285 , : 8522. ( ) : 982 , : 203
( ) : 9832 , : 2039
( ) : 98320 , : 20389
3. ( ) ( )
( ) ( )
4. ( ) ( )
( ) ( )
( )
5. ( ) 10, 40, 9, 5, 5,
0, 12 ( )
6. = 10 , = 25 , = 0 , = 192 , = 16 < < < < 7. 39, 93, 48, 84, 57, 75 , 66
8. ( ) 26,65 ( ) 10
( ) 7,82 ( ) 16,328 9. ( ) 140 ( ) 35,28
( ) 778 ( ) 184 ( ) 5148 ( ) 18,6
( ) 7,875 ( ) 18
10. ( ) 5700 ( ) 8200
( ) 46000 ( ) 0,09
( ) 110 ( ) 200011. ( ) 81 243 – 729 ( ) 83 65 47
( ) 6,5 7,1 7,7 ( ) 800 400 200
12. ( ) 100 ( ) 0,0574
( ) 0,01 ( ) 0,00001
( ) 57400 ( ) 0,53
192
:
13. 0,045
14. ( ) , , ,
( ) , , ,
( ) , , ,
15. = 806600 = 80,66
= 8066 = 8066
16. ( )
i. 8 ii. 8 iii. 3 iv.
17. ( )
i. ii. iii. iv.
( ) 5654 ( ) 6120
( ) 37962
18. ( ) , ( )
17
1. ( ) > ( ) <
( ) = ( ) <
2. : : : : 524
3. ( ) 2 ( )
( ) 3 ( ) 1
( ) 8 ( )
( ) ( )
( ) 10 ( )
( ) ( ) 5
4. 171000000 5. + + + +
6. = 0,9 = 1,6 = 2,5 = 0,56257. 20
8.
9. ( )
193
:
10. : : :
1
1. ( ) = 20 , = 25
( ) = 26 , = 40
( ) = 20 , = 20
( ) = 18 , = 14
2. 1800
3. ( ) ( )
( )
4. 8
5. 140
6. 1
7. ( ) 135,2 ( ) €500,24
29
1.
= { , , , , }, ( ) = 5 = {86, 88, 90, 92, 94, 96, 98} ( ) = 7 = { } , ( ) = 0 = {18, 27, 36, 45, 54, 63, 72, 81, 90, 99} ( ) = 10 = { , , , , , , } ( ) = 7
2.
: : 1,2,3 : 11
3. ( ) ( )
( ) ( )
( ) ( )
( ) ( )
5. , , , ,
194
:
37
1. ( ) = {3, 4, 5, 6, 7, 8} ( ) = {4, 6}
2.
3. ( ) = { , , , , , , , , , , } ( ) = { , , } ( ) = { , , } ( ) = { , , , , , , , , , }
4. ( )
( )
( )
5. = { , , , , } 7. ( ) . . { , , , } { , , , , } = { , , }
( ) {22 ,33 , 11} {11 , 22, 55, 33 , 66} = {11, 22, 33, 55, 66} 8. 8
9. ( )
3
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3. ( ) 5, 3 1,2 ( ) , , ,
4. . . = {1,2} = { , #} 5. ( ) . . = {2,4,6,8, … } ( ) . . = {1,2,3,4,5} 6. ( ) ( )
195
:
( )
1. = { , , , } ( ) = 4 = { } ( ) = 0
= {7,0,1} ( ) = 3 = { , } ( ) = 2 = { , , , , , , } ( ) = 7
2.
: . . : . : 21 : : 224466 :
3.
4. ( ) ( )
( )
5. ( ) = { , , , , , , , , } ( ) = { , , , , } ( ) = { , , , , , , , , } ( ) = { , , }
7. ( ) , , , , , ,
8. ( ) iv ( ) iii
( ) i ( ) ii
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196
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( ) ( ) = ( ) + ( ) ( )
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6. ( ) ( )
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1. ( ) 3 ( )
( ) 2013 ( )
( ) ( )
2. ( ) 49 : 7 : 2 ( ) 144 : 12 : 2
( ) 8 :2 : 3 ( ) 10000 : 10 : 4
( ) 243 : 3 : 5 ( ) 1 : 1 : 3
( )1234 : 1234 : 1
( ) 1 : 5 : 0
( ) 1 : 2012 : 0
3. ( ) 2 ( ) 7
( ) ( ) 10
4. ( ) . . 1 , 1 , 2014 , …
( ) 8 4 2 ( ) 9 3 81
( ) 16 4 2
5. ( ) 390625 ( ) 20736
( ) 117649 ( ) 4096
6. 1 < 5 < 2 < 5
7. 5
8.
: 21 2 = 1048576 €10484,76 20 €5242,38, . . . ,
€15000 .
9. ( ) 8 ( ) 29
197
:
62
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( ) 5 ( ) 8
( ) 8 + 1 ( ) 13 5
2. ( ) ( )
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3. ( ) 8 ( ) 2
( ) 129 ( ) 21
( ) 10 ( ) 81
10. ( ) 14 ( ) 5
( ) 148 ( ) 5
( ) 112 ( ) 26 11. ( ) 2 (5 + 2) = 11 23
( ) 22 (7 5) 2 = 2 3 4 12. 6, 2, 5, 3
57
1. ( ) 13333312. 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010
3. ( ) 6 ( ) 15
( ) 9 ( ) 127
( ) 113 ( ) 170
4. ( ) 1010 ( ) 1000
( ) 1110 ( ) 10001
( ) 1110 ( ) 101111
( ) 110100 ( ) 1000011 5. ( ) 100, 101, 110
( ) 1110, 1111, 10000 ( ) 1100, 1101, 1110
6. ( ) : 100 : 111 ( ) : 4 : 7
7. : 8, 9, 10, 11, 12, 13, 14, 15
198
:
( ) 5 ( ) , 10
( ) 2 ( ) 3
4. ( ) = 4 ( ) = 6
( ) = 3
5. , ,
6. ( ) + 2 ( ) €14
( ) 2 ( )
( ) 3 4 ( ) 5( + 2)
( ) €20
7. = 4 + 3
8. ( ) : 3 :
( ) : 3 + 7 : + 7
( ) : 3 6 : 6
9. ( )
10. , , , ,
67
1. ( ) = ( ),
( ) i. = 44 ii. = 77
2. ( ) 36 ( ) 9
( ) 12 ( ) 0
( ) 26 ( ) 29
3. ( ) : 8, 10, 13, 15, 20
: 44, 54, 69, 79, 104 ( ) = 5 + 4
4. ( ) 22 ( ) 42
5. ( ) = 20 ( ) = 4
( ) 32
6. = {5,10,15, … } = {1,3,5,7, … } = {5,9,13,17,21}
7.
= {2 } = { + 1 } = { } = { | < 5}
199
:
8. ( ) : : = 100 + + 2 ( ) = 357 €3,57
– 72
1. ( ) ( ) 2. ( ) ( )
( ) ( ) ( )
3. ( ) = 9 + ( ) = ( ) = 4 ( ) = 13 + ( ) =
4. ( )
( ) 6. 2012, 2013, 2014
H 78
1. 10 , 11
2. ( ) = 7 ( ) = 30
( ) = 15 ( ) = 64
( ) = 3125 ( ) = 510
( ) = 57 ( ) = 5
( ) = 5 ( ) = 15
3. ( ) = 11 ( ) = 7
( ) = 3 ( ) = 6
( ) = 3 ( ) = 25
( ) = 15 ( ) = 2
4.
+ = = = = + = = = = = = = =
5. . . + 5 = 15, 5 = 50, + 2012 = 2022,
100 = 10, = 9 , …
200
:
6. ,
. 7. 2 + 3 = 118. 22
9. €13680
10. 10
11. €12
12. : 6 12
13. 14 16
14. : 2 : 8
15. 30
16. = 12 , = 5 , = 13
17. ( ) 24
( ) ,
18. = 10
81
1. 2
2. ( ) = 84 ( ) = 24
( ) = 28 ( ) = 110
( ) = 24 ( ) = 11
( ) = 5 ( ) = 9
3. ( ) = ( ) >
( ) = ( ) =
( ) < ( ) >
4. ( ) 3 9 ( ) 1000 10
( ) 3 27 ( ) 36 6
( ) 12 144 ( ) 2 4 16
5. ( ) 122 ( ) 0
( ) 36 ( ) 13
( ) 1 ( ) 17 ( ) 20 ( ) 134
6. = 17, = 27, = 7
201
:
7. ( ) 3 ( ) 6
( ) 15 ( ) 18
8. ( ) 111 ( ) 10010
( ) 1110101 ( ) 10110101
9. ( ) 6 ( ) 2 + 4
10. ( ) ( )
( ) ( )
11. ( ) = + 600 ( ) = ( + 600)
( ) = + 1080
12. ( ) , 6 = 11( ) , 4 = 20
( ) , + 5 = 8
( ) , 7 = 12
( ) , 6 = 54
( ) , 4 + 12 = 26
13. 3, 9, 27, 81, … 14. : 36 : 12
15. 194, 195
17. 242, 243, 244
18. = 16 , : 4 , = 16
19. = 26, = 40
20. €4
21. ( ) 1001, 1010, 1100 ( ) 9, 10, 12
8
1. ( ) o , 4 + 1 6 , 25 .
2. . .
4. 3. 28
4. ( )
( ) 12 = 1100( ) = 14( )123 = 1111011( ) = 173( )
202
:
724 = 1011010100( ) = 1324( ) ( ) . i. ii.
( ) . i. 736( ) = 1050( ) . ii. 9
5. ( ) 11110110( ) ( ) 10011111( ) ( ) 1100100( )
8. ( ) 580
( ) 2 1
3
93
1. ( ) = 19 , = 0 ( ) = 30 , = 2
( ) = 42, = 0
2. = 7 , = 4
3.
739, 100, 20, 15 4. 36
5. ( ) ( )
( ) ( )
( ) ( )
( )
6. 4
7. 19, 38
8.
9. ( ) ( )
10. ( ) 57 ( ) 36
11. 2 ,2 , 9
203
:
97
1. ( )
( )
( )
( )
3. ( ) 2814 28028
( ) . . 1428, 4242, …
4.
6. ( ) = 2 + = 2 +
7. 14, 10, 7, 5, 2
8. = 12, 24, 36, ….
1
1. ( ) 1520, 4404, 3850 ( ) 765, 4404
( ) 1520, 4404 ( ) 765, 1520, 3850
( ) 3850 ( ) 765
2. ( ) 0 2 4 6 8 ( ) 2 6
( ) 1 4 7 ( ) 6
( ) 2 8
3. ( ) 2 3 4 6 7 8 9 ( ) 0 2 3 6 8
( ) 0 1 2 3 4 5 8 9
( ) 0 ( ) 4 5
4. ( ) 100 ( ) 100
( ) 102 ( ) 108
( ) 100 ( ) 105
5.
6. 0 3 9
7. 90
8. 102
9. 6
10. : 64 : 90
14. ( ) 37 , 111, , 3 , 37
204
:
11
1. 56 = 2 7 , 108 = 2 3 , 73 = 73 , 84 = 2 3 7 , 65 = 5 13
: 73 2. 42
3. 30
4. . . 2 + 3 = 5
5. ( ) 32 = 3 + 29
( ) 20 , 23, 29, 31, 37, 40
6. 211
7.
11. 28
12. ( ) 12 = 3 4 3|(2 3 5 7 )4|(2 3 5 7 ) 12|(2 3 5 7 )
( )
15 = 3 5 3|(2 3 5 7 )5|(2 3 5 7 ) 15|(2 3 5 7 )
( )
21 = 3 7 3|(2 3 5 7 )7|(2 3 5 7 ) 21|(2 3 5 7 )
13.
116
1.
(32, 36) = 4, [32, 36] = 288 (24, 45) = 3, [24, 45] = 360 (32,48,80) = 16, [32, 48, 80] = 480 (252,294, 378) = 42, [252,294, 378] = 5292 2. , 4
3. ( ) ( )
( ) ( )
4. ( ) ( )
205
:
( )
5. ( ) ( )
( )
6. 18 , : 6, : 1, : 3
7. 150 , : 3, : 5
8. 24 , 1
9. 45
10. . . 5, 9
11. = 24
12. = 13 = 26 = 52
13. 14 35
15. , 3, 6
16. 4
119
1. ( ) 195 ( ) 42
( ) 66
2. ( ) ( )
( ) ( )
( ) ( )
( )
3. 180, 270, 360, 450, 540, 630, 720, 810, 900, 990
4. 2184
5.
6. 36
7. 12 13 15 17 21 31 51 71
9. 78
10. 1080
11. 8 4 , 6 , 9
12. 252
13. 2010
14. €195
206
:
121
1. ( ) 8 ( ) 24
2. 2520
3. . . 4, 9, 10
4. 501
–
– 13
3.
2013 99 4,052 +0,023
3
4. ( ) ( )
( ) ( )
( ) ( )
5. ( ) 1 ( ) . . 3, 1
( )
6. ( ) 2 ( ) 7,5
( ) 12 ( ) ±8,3
( ) 0 ( ) ±123
7. ( ) 2 ( ) 15
( ) 11
8. ( ) 2, 1, 0, 1, 2 ( ) 0
( ) 2, 1, 0, 1, 2
9.
207
:
13
1. ( ) < ( ) <
( ) < ( ) <
( ) < ( ) >
( ) > ( ) >
( ) > ( ) <
2. 280,21 280,12 28,12 28,012 2,8
3. . . 107 < 62, 62 < 23, 15 > 234. ( ) . . +8,1 > 5 ( )
( ) ( ) . . | 5,999| < +6
( ) ( ) . . 16 > | 16| 5. ( ) ( )
( ) ( )
6. ( ) . . 2 < 1 < 0 < < 1
( ) . . < < 0 < 1 < 1
7. ( ) > ( ) >
1 1
1. ( ) 1 ( )
( ) 1 ( )
( ) 0,45 ( ) 6,55
2. ( ) +22 ( ) +22
( ) +6 ( ) 22
( ) 28 ( ) 30
( ) 13 ( ) 54
( ) 0 ( ) 0
3. ( ) 4,15 ( ) 0,8
( ) + ( ) 5
( ) 2,35 ( ) +3
( ) 1 ( ) 1
4. ( ) +7 4 = +3 ( ) 3 + 2 = 1
208
:
( ) 3 2 = 5
6. ( ) +8 ( ) 10,1 ( ) +20 ( ) +40
( ) +7 ( ) 3
( ) 4
7. +3
8. ( ) 9 ( ) 14
( ) 13 ( ) 1
( ) 11 ( ) 3
9. +3 4 +1 2 0 +2 1 +4 3
1 6 +1 0 2 4 5 +2 3
10. 38,92
11. = 0,9 = 5,3 = 6,2
12. ( ) ( )
( ) ( )
13.
14. +9 6 3 = 7 + 5 + 2 = 5 + 1 + 4
15. ( ) 9 ( ) 9 ( ) 11 2 ( ) 9 14
( ) 9 + 2 ( ) 2 + 3 3
16. ( ) = = 0
( ) = 0
( ) = = 0
1 7
1. ( ) 30 ( ) +35
( ) 2 ( ) +23
( ) 25 ( ) 0
( ) +25 ( ) +18
2. ( ) 1,4 ( ) +6,8
( ) 15,2 ( ) 1
209
:
( ) +29 ( ) +2
3. ( ) 2, 0, 2, ( ) 2, 5, 8 ( ) 1, 6, 11 ( ) 7 , 9, 10 ,
4. ( ) < ( ) <
( ) > ( ) >
( ) < ( ) <
5. 53
6. ( ) 4 ( ) +15
( ) +18 ( ) 24
( ) +9 ( ) +13
( ) ( ) 1
( ) +2
( ) +7
7. ( ) 14 ( ) +2
( ) 12 ( ) +7,4 ( ) +
8. = 2 , = 3
9. ( )
( ) : 2 ( 1) = 1
( ) : +2 (+11) = 9
( )
10. A = 10,8 11, B = 87,52 88
11. ( ) ( ) +,
( ) , +,
12. ( ) .
( ) 17 00
155
1. ( ) 1 ( )
( ) ( ) 16,32
2. ( ) 14 ( ) +10
( ) 0 ( ) 6
( ) 36 ( ) +55
210
:
( ) +18,6 ( ) 0
( ) 4 ( ) 6,72
( ) ( ) +
3. ( ) 6 ( ) +6
( ) 12 ( ) 6
( ) 0 ( ) +1
( ) + ( ) +1
( ) 0 ( ) +1
4.
+3 7 2,5 +1
3 +7 +2,5 + 1
5. ( ) 16, 32, +64
( ) 2, 2, 2
( ) 10000, 100000, 1000000 ( ) 81, 243, 729
6. ( ) ( )
( ) ( )
( )
7. ( ) +6 ( ) 15
( ) 12 ( ) 4
( ) +7 ( ) 36
( ) 47 ( ) +7
( ) +3 ( ) 0
8. ( ) +1 ( ) +1
( ) +1 ( ) +
( ) 0 ( ) 1
( ) 1 ( ) 9.
. . ( 1)(+6) , (+1)( 6), ( 2)(+3) , (+2)( 3) (+1)( 2)(+3), (+1)( 1)(+1)(+6), ( 1)( 2)( 1)( 1)( 3), ( 1)(+1)( 1)( 1)( 1)( 6) …10. ( ) ( 2100) ( 2,5)
( ) ( 2100) (+10)
11. = 2, = +1, = 0, = +10
12. ( ) 3 6 ( ) 3 + 8
211
:
( ) 3 + 16 ( ) 18
( ) 22 10 ( ) 2
( ) + 4 ( ) 3y + 55
13. ( ) i
.i. 350 ii. 6
iii. 9(350 ) iv. 3 + 3150
( ) €2790
14. 2
16
1.
( 2) 2 3 ( 2)( 2)( 2) ( 6,3) 6,3 2 ( 6,3)( 6,3)
13 4
2. ( ) ( )
( ) ( )
3. ( ) + ( ) +
( ) + ( )
( ) ( )
( ) + ( )
( ) +
4. ( ) 25 ( ) 16
( ) ( ) ( ) 1 ( ) 1
( ) 27 ( ) 100
5. ( ) 25 = ( 5) = (+5) = (+25)
( ) 8 = ( 2) = ( 8)
( ) . . 1 = ( 1) = (+1) = ( 2012)
6. ( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
212
:
7. ( ) = ( ) =
( ) > ( ) =
( ) < ( ) >
8. ( 6) < 6 < ( 6) < 6
9. : 0 , 1 : 0 , 10
: 0 1 10. = 16
11. = 0
165
1. = 5, = 4 , = +1 , = , = , = + < < < < <
2. ( ) +5 ( ) 3
( ) 6 ( ) +10 ( ) +0,7 ( ) 0,5
( ) 100 ( ) +3,06
( ) 40 ( )
( ) ( ) +98
3. ±1 , ±5
4. . . 4, +2, +6
5. ( ) < ( ) >
( ) < ( ) =
( ) > ( ) >
6. ( ) +4 ( ) +4
( ) 5 ( ) +12
( ) +12 ( ) +25
( ) + ( ) 3
( ) 8 ( ) +2
( ) 24 ( ) . . …
7. : 45 : 50
8. ( )
( )
( )
213
:
9. ( ) +4 ( ) +1
( ) 0 ( ) +4
( ) ( ) +
( ) ( ) +
10. ( ) 12 ( ) 19
( ) +9
( ) ( 20) [2 ( 2)], [( 21) + 6] 3, 10 + 3 (2 7)
11. ( ) ( 10) ( 2) + (+1) = 21 ( ) ( 5) ( 2) + (+4) = 1
( ) 6 ( 7) (+2) = 44 ( ) ( 8) ( 2) ( 2) = 9 12. ( ) 38 ( ) +19
( ) 6 ( ) +102
( ) 0 ( ) +4
( ) 1 ( ) 2 ( ) +7 ( ) 2
( ) ( )
13. ( ) 3 ( ) 2
( ) 3 ( ) 160
( ) 30 ( ) 7
14. ( 5) + [(+4) ( 2) ( 1)] ( 3)
15. ( ) ( )
( ) ( )
( )
16. ( ) +9 ( ) 19
17. ( ) 27 ( ) 7
( ) 16 ( ) 3
( ) 2 ( ) 4
( ) 31 ( )
172
1. ( ) ( )
( ) ( )
2. 1, 27 , 0,375 , 0,83
3. ( ) 1, 1 > 1,11111111 ( ) 3,68 > 3,68
214
:
( ) 4,912 > 4, 102
4. 0,069 0, 6 0,69 0, 69 0,69
178
1. ( ) 2 ( ) 6
2. ( ) = 12 ( ) = 4
( ) = 3 ( ) = 5
( ) = 3 ( ) = 1
3. ( ) = 2 ( ) =
( ) = 19 ( ) = 2
( ) = 10 ( ) = 2
4. ( ) = 3 ( ) = 6
( ) = 2 ( ) = 3
( ) = 9 ( ) =
( ) = 8 ( ) = 2
6. : 5
7. : 57
8. : 7 cm , : 34 cm , : 10 cm , : 35 cm
9. : 140 , : 80, : 60
10. = 16 cm = 15 cm
11. : €19 : €7
13. : 10
14. 43
181
1. ( ) +6 ( ) 5
( ) +1 ( ) 5
( ) +6 ( ) 4
( ) +2 ( ) +60 ( ) +2 ( ) +5
215
:
2. ( ) + ( ) +
( ) + ( ) +
( ) ( )
3. ( ) ( )
( ) ( )
( ) ( )
( )
4. ( ) 9 ( ) 1
( ) 16 ( ) 1
( ) ( ) 16
( ) 1 ( ) 25
( ) 1 ( ) 0
5. ( ) 1,5 ( ) 4, 3
( ) 1, 81 ( ) 0,875
6. ( ) > ( ) >
( ) < ( ) >
( ) > ( ) >
( ) > ( ) <
( ) < ( ) <
( ) > ( ) =
7.
8. ( ) 9 ( ) +6,8
( ) 5 ( ) 8
( ) ( ) +8
( ) +5 ( ) 5
( ) 3 ( ) 4
( ) +2 ( ) 0
9. ( )
( ) : 2 > 7 , | 2| < | 7| ( ) : |3 ( 1)| |3| | 1| ( ) : |3 ( 1)| 3 ( 1)
( )
10.
11.
216
:
12. ( ) +12 ( ) +4
( ) +14 ( ) 17
( ) 0 ( ) 20
( ) +8 ( ) 6
( ) +10 ( ) 14
( ) +1 ( ) +
13. iii.
14. ( ) = 5 ( ) = ±12 ( ) = 0 , = ±1 , = ±2
15. ( ) 0 ( )
( ) +2 ( ) 33
( ) 19 ( ) 4
16. ( ) ( )
( ) ( )
( ) ( )
( )
17. A , i , ii , iii
18. ( ) > ( ) <
( ) < ( ) <
( ) < ( ) >
19. ( ) ( )
( ) ( )
21. = 26 = +7
22. ( ) +9 ( ) 25
23. ( ) = 2 ( ) = +3
( ) = 6 ( ) = 1
( ) = +1 ( ) = 2
( ) = 10 ( ) = +7
( ) = 6 ( ) = +2
24. 121
25. : 33 : 6
26. : 80 : 20 : 15
27. : €60
: €20 : €20
: €60
217
:
28. : 10 : 40
29. : 30
30. : 36 : 12
16
1. 17
2. < 0
3.
4. : 30 10
5. 0
6. ( ) : 16 : 18
( ) 6
7. ,
8.
9. = 3
10. 4
11. 32
12. 19683
{ } =
{1,2,3,4, . . . } {0,1,2,3,4, . . . } {0, ±1, ±2, ±3, ±4, . . . } {1,2,3,4, . . . } { 1, 2, 3, 4, . . . } { / : ,
/
o
=
·
Ans
^
SHIFT 2nd F
AC
DEL
Abs
(-)
a b/c 23
S D a b/c
CLR
M+
M-
M
3 + 4 3+4 7 2,34 – 1,1 2.34 – 1.1 1.243 2 3 2 6 2 2 2 32 5 10 5 3 5 103 5000
3 4 2
2
2 3 4 4 4 2 5 5 25 2 (7 3) 2 (7 3) 8 2 (7 3) 2 8 2 ( 3) 2 ( 3) 5 | 3| | 3| 3 0,25
9 6 2
9 – 6 2 + 6
3 (9 6: 2) 6: (9 6: 2) 3 – 6 17
+ = 3 4
· – 2 3 4 1 · 1 =
3 x 2 =
2 5 =
5 P 3 =
3 a b/c 4 =
3 a b/c 4 =
SHIFT 23 2 3 4 =
2 a b/c 3 a b/c 4 =
4 =
5 X² =
= 3 – 7 ( x 2 )
x 2 = 3 – = 7 Ans
2 ( (-) 3 ) =–
= 3 (-) Abs
S D
a b/c SHIFT
= 2 ÷ 6 – 9
3 – 6 ÷ =